Tecnam P92 Instructor reported engine roughness resulted in a safe off airport landing. Instructor suspected fuel contamination.

Narrative

Skies were clear; outside air temperature approx 45 degrees Fahrenheit on the ground; temperature approx 40 degrees Fahrenheit at 2500 ft. MSL. Practicing maneuvers in the air; steep turns; slow flight; power on stalls; power off stalls; in that order; on the recovery from the second power off stall (dropping the nose of the airplane to break the stall; and introducing full power) aircraft immediately starting shaking violently and the engine was running extremely rough; at full power the engine was producing 3000 RPM and intermittently jumping to approx. 3500 RPM (Normal full power produces around 5200 RPM) attempted different throttle settings to try and find a setting that would produce enough power to maintain altitude; no improvements. Engine instruments were all reading green or in the normal range; ran checklist to ensure the engine was getting fuel and no improvement on the performance or vibrations from engine; turned immediately towards the closest Airport ZZZ and communicated on the radio with UNICOM to tell them we probably weren't going to make it to the Airport; reported on 121.5 (GARD) with tail number and location and problems we were having; could not hold altitude from 2500 ft. MSL. Made the decision to land in a field to avoid overflying trees; flew a rectangle around the field to land into the wind. Landed in the field approximately 11 miles N/NE of ZZZ; communicated with cell phone to appropriate people (FBO; FSS; ZZZ TRACON; Airport Manager; Maintenance) no visible damage to aircraft and no injuries to myself or my student. I believe there may have been impurities in the fuel which would have caused the engine to run improperly or one or both of the carburetors developed ice (no carb heat in the plane) or water in the fuel lines may have frozen or one of the carburetors could have been faulty.